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Chapter 32: Problem 43

What order is necessary to resolve 647.98 -nm and 648.07 -nm spectral lines using a 4500-line grating?

Short Answer

Expert verified
The diffraction order necessary to resolve the spectral lines 647.98 nm and 648.07 nm using a 4500-line grating is approximately 2.

Step by step solution

01

Understand the Grating Equation

The grating equation is given by \( mλ = d \sin θ \) where,\n\n- \( m \) is the order of diffraction\n- \( λ \) is the wavelength of light\n- \( d \) is the spacing between the slits in the grating\n- \( θ \) is the diffraction angle\n\nThe problem gives us the wavelengths and the total number of lines in the grating. We can calculate the order of diffraction thus.
02

Calculate the Difference in Wavelengths

The first step is to calculate the difference in the wavelengths, that is, \( Δλ = λ2 - λ1 = 648.07 - 647.98 = 0.09 \) nm. Note that the values need to be converted to meters, so \( Δλ = 0.09 * 10^{-9} \) m.
03

Calculate the Spacing

Grating spacing is calculated by \( d = \frac{1}{N} \) where \( N \) is the total number of lines in the grating. Substituting \( N = 4500 \), we get \( d = \frac{1}{4500} \) m.
04

Apply the Rayleigh Criterion

From the Rayleigh Criterion, \( Δλ = \frac{λ}{mN} \). We know the values for \( Δλ \) and \( N \). We'll take the average of the wavelengths given, i.e., \( λ = \frac{647.98+648.07}{2} = 648.025 \) nm or \( 648.025 * 10^{-9} \) m. Substituting all the known values we can calculate the order \( m \).

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